Taking a screenshot is convenient, but it is usually not the best way to save a wallpaper. A screenshot captures what is currently displayed on your screen—including any scaling, interface overlays and the resolution limits of that display.
A screenshot is a copy of the screen, not the original image
When an image is displayed inside a social app, the app may resize it to fit the screen. A screenshot records that resized version. If the original image was larger than your display, the extra detail is not preserved in the screenshot.
Direct downloads avoid interface elements
A downloaded image does not include captions, buttons, status bars or other app controls. That makes it easier to set the image directly as a lock-screen or home-screen background without additional cropping.
Screenshots can introduce another generation of processing
The screenshot itself is encoded as a new image file. Modern phones handle this well, but you are still creating a new copy rather than keeping the source file that was made available for download.
When screenshots are still useful
Screenshots are useful for references, temporary mood boards, layout ideas or cases where you intentionally want only a portion of an image. They are simply less suitable when your goal is to preserve the highest available quality of a wallpaper.